J. Fluid Mech. (2020), vol. 892, A19. c© The Author(s), 2020.Published by Cambridge University PressThis is an Open Access article, distributed under the terms of the Creative Commons Attributionlicence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, andreproduction in any medium, provided the original work is properly cited.doi:10.1017/jfm.2020.181

892 A19-1

Minuet motion of a pair of capsules interactingin simple shear flow

X.-Q. Hu1,2, X.-C. Lei2, A.-V. Salsac1,† and D. Barthès-Biesel1

1Biomechanics and Bioengineering Laboratory (UMR CNRS 7338), Université de Technologie deCompiègne, Alliance Sorbonne Université, CS 60319, 60203 Compiègne, France

2State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College ofMechanical and Vehicle Engineering, Hunan University, 410082 Changsha, China

(Received 29 September 2019; revised 22 February 2020; accepted 28 February 2020)

We study the three-dimensional hydrodynamic interaction of a pair of identical,initially spherical capsules freely suspended in a simple shear flow under Stokes flowconditions. The capsules are filled with a Newtonian liquid (same density and viscosityas the suspending fluid). Their membranes satisfy the neo-Hookean constitutive law.We consider the rarely studied case where the capsule centres are initially locatedin (or near) the plane defined by the flow direction and the vorticity vector, i.e. intwo different shear planes. The motion and deformation of the capsules are modelledby means of a boundary integral technique to compute the flows, coupled to a finiteelement method to calculate the force exerted by the membranes on the fluids. Wefollow the motion and deformation of the capsules as they are convected towardseach other after a sudden start of the flow. Our main finding is that, depending ontheir initial position and deformability, the two capsules may oscillate slowly aboutthe flow gradient axis, get nearer to each other at each oscillation to finally interactstrongly and separate. This minuet motion had not been identified previously. Weidentify the regions of space where either simple crossing or minuet occurs. Thisphenomenon has a marked influence on the irreversible trajectory drift of two capsulesafter crossing: the minuet process leads to a significant trajectory displacement alongthe flow gradient when none was expected, based on the previous studies where thetwo capsules had a significant relative velocity.

Key words: capsule/cell dynamics, suspensions, boundary integral methods

1. IntroductionThe hydrodynamics of pairwise interaction of deformable particles is a crucial topic

for semi-dilute suspension rheology (Batchelor & Green 1972a; Guazzelli & Morris2012). When the particles are deformable, their shear induced deformation leads tonon-Newtonian and to self-diffusion effects. This has been demonstrated for liquiddroplets (Loewenberg & Hinch 1997; Guido & Simeone 1998). The case of capsules

† Email address for correspondence: anne-virginie.salsac@utc.fr

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892 A19-2 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

(liquid drop enclosed by a thin elastic membrane) is particularly complex becausethe motion and deformation of those particles result from nonlinear fluid–structureinteractions that are difficult to model. For example, the capsules may be highlydeformed as they cross each other, which leads to the formation of a thin liquidfilm between the two particles and to potential damage of the membrane due to highshearing forces.

This phenomenon is usually studied in a simple shear flow with velocitycomponents given by v∞1 = γ x2, v

∞

2 = v∞

3 = 0 in a laboratory Cartesian referenceframe, where γ is the shear rate. The two capsules C1 and C2 are initially positionedwith distances 1X(0)

1 , 1X(0)2 , 1X(0)

3 between their centres. The first three-dimensionalmodel of two initially spherical identical capsules (radius a) interacting in simpleshear flow is due to Lac, Morel & Barthès-Biesel (2007), who considered the casewhere the two capsules had their centres in the same x1x2 shear plane (1X(0)

3 =0). Thecapsule membrane is treated as a very thin sheet of a hyperelastic material devoidof bending resistance and the flow Reynolds number is assumed to be negligible.Lac et al. showed that, in a reference frame centred on C1, capsule C2 is firstdisplaced along the velocity gradient so that it can overpass (‘jump over’) C1 andit is then shifted back towards the flow axis as it moves away. However, the finalseparation 1X( f )

2 is larger than the initial one 1X(0)2 . The crossing thus leads to an

irreversible trajectory shift along the shear gradient (x2-direction). This effect, whichdecreases with an increase of the capsule deformability and/or the initial distance1X(0)

2 , ultimately leads to self-diffusion effects in a suspension. We propose to callthis crossing process the leapfrog motion. The same situation was later consideredwhere the two spherical capsules were replaced by two red blood cells (Omoriet al. 2013) or two vesicles (Gires et al. 2014). In both instances, it is found thatthe particles do a leapfrog motion with a trajectory shift that evolves qualitatively,as found previously by Lac et al. Experimental measurements of the trajectory ofliquid filled giant lipid vesicles compare well with the predictions of the flow model(Kantsler, Segre & Steinberg 2008; Gires et al. 2014). In all the aforementionedstudies, the flow field around the capsules was computed by means of the boundaryintegral representation of the Stokes equations. As a consequence, the pair of capsulesis effectively interacting in an infinite flow domain.

Finite differences and front tracking techniques can also be used to study theinteraction problem: the advantage is that non-Newtonian or finite inertia effects inthe suspending fluid can be considered. However, the computation is then usuallyperformed in a flow domain, bounded by two walls parallel to the x1x3-plane wherethe velocity is imposed (to create the simple shear flow). On the other boundariesof the box, periodic flow conditions are imposed. Doddi & Bagchi (2008) usedthis technique to model the pair interaction of two initially spherical capsules,when the inertia of the flow was not negligible. They found that, when the flowReynolds number increased, the capsules did not cross, but reversed their motion.This phenomenon was confirmed for a pair of liquid droplets (Olapade, Singh& Sarkar 2009). However, the spiralling motion reported by Doddi and Bagchi,is linked to the size of their computational domain and is a confinement effect.Indeed, for the results to be independent of the computational procedure and tobe transposable to unbounded flow situations, the domain has to be large enough,typically 40a × 10a × 5a, for in-shear-plane crossing (Olapade et al. 2009). Pranayet al. (2010) considered the case where the suspending fluid is a dilute polymericsolution. They found that the presence of polymer leads to a decrease of the trajectoryshift only when the capsule deformability is low. A recent study (Singh & Sarkar2015) considered the pair interaction of two capsules with different deformability

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Minuet motion of two capsules 892 A19-3

and found that the trajectory shift is influenced by the stiffness ratio. All the studiesbased on finite difference and front tracking techniques only considered pairs ofcapsules with their centroids in the same shear plane, where they remain because ofthe problem symmetry.

However, in a suspension, two nearby capsules will not necessarily have theircentres of mass in the same shear plane. For example, special care has to be takenin vesicle experiments to ensure that this is approximately the case. Accordingly,Lac & Barthès-Biesel (2008) also modelled the three-dimensional motion of twocapsules positioned in two different shear planes (1X(0)

3 6= 0). In this case, a sidewaysleapfrog motion occurs with a maximum trajectory displacement along both the x2-and x3-directions, which decreases as 1X(0)

3 and/or capsule deformability increase.Lac and Barthès-Biesel also showed that the final trajectory shift along the x3-axisis indeed smaller than the one along the velocity gradient x2-axis (approximatelyone third), but present nevertheless. This was also globally confirmed by Gires et al.(2014) for vesicles.

However, in all aforementioned studies, the two capsules always had a significantinitial relative velocity, obtained by means of 1X(0)

2 > 0.5a. This choice was madeto avoid very long computations. The only exceptions are due to Lac et al. (2007)and Omori et al. (2013), who showed that, for two capsules located on the samestreamline (1X(0)

1 6= 0, 1X(0)2 =1X(0)

3 = 0), the perturbation created by the deformationand membrane rotation of the two capsules created a small but finite velocity field thatdisplaced the centroids along the x2-axis and led to a relative velocity of the capsulesand to a leapfrog motion. The result was interesting as it showed that such initialconditions led to the largest self-diffusion effect. The conclusion of this review is thatthere is presently no information on the interaction of two capsules when their centresare located in the x1x3-plane.

The objective of this paper is to fill this gap and to investigate the three-dimensionalmotion of two capsules when their centroids are in or near the same x1x3-plane. Wewill take advantage of the computational technique that we have developed, basedon the coupling of a boundary integral to compute the flow and finite elements tocompute the capsule wall mechanics (Walter et al. 2010). This coupling has provedto be very stable in a number of situations where long transient motion of a singlecapsule needed to be monitored (Walter, Salsac & Barthès-Biesel 2011; Hu, Salsac& Barthès-Biesel 2012; Dupont, Salsac & Barthès-Biesel 2013; Dupont et al. 2016).We will see that a new interaction mode is revealed: given the choice, the capsulesoscillate around the shear gradient axis, rather than around the vorticity axis. We callthis interaction mode the minuet motion, as it is similar to the one reported for Volvoxalgae (Drescher et al. 2009), albeit for different hydrodynamic interactions.

The paper is organized as follows: the problem is set out in § 2 together with ashort description of the numerical method. The different types of capsule interactionare presented in § 3, where we also discuss the main factors that determine the motiontype. In § 4, we analyse which factors determine the motion type and illustrate thearea of space where oscillatory motion is expected to occur. In § 5, we then study theconsequences on the trajectory shift and self-diffusion phenomena in the suspension.In the final § 6, we summarize the findings and provide a conclusion.

2. Problem statement and numerical method2.1. Problem description

Two identical spherical capsules C1 and C2 (radius a), filled with a Newtonian liquid(viscosity µ, density ρ) and enclosed by a very thin hyper-elastic membrane (surface

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892 A19-4 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

x2

x1

x3Shear plane

Vorticitydirection

Initial position{X1

(0), X2(0), X3

(0)}

FIGURE 1. Two capsules flowing in simple shear flow. The reference frame is linkedto capsule C1 and the coordinate system is centred on it. The centre of C2 is initiallypositioned at X(0).

shear modulus Gs and area dilation modulus Ks), are freely suspended in anotherNewtonian liquid (viscosity µ, density ρ) and subjected to a simple shear flow withshear rate γ . Inertia effect is neglected. The capsules centroids are denoted G1 and G2.We use a reference frame centred on G1, that moves with it (figure 1). Our objectiveis to study the interaction process between the two capsules as they are convected bythe flow and, specifically, to compute the evolution of the velocity V(t) and positionX(t) of G2 with time t.

The undisturbed flow v∞ of the external liquid is given by

v∞1 (x)= γ x2; v∞2 (x)= v∞

3 (x)= 0. (2.1a,b)

Note that, since the problem is inertialess, the flow field has the same expression(2.1) in a laboratory reference frame. The motion of the internal and external fluidsis governed by the Stokes equations, with associated boundary conditions given by:

(i) vanishing flow perturbation far from the capsules:

vext(x)→ v∞(x) as ‖x− x(Gα)‖→∞ α = 1, 2; (2.2)

(ii) at a material point x located on the membrane of either capsule

vext(x)= vint(x)= x, (2.3)q+∇s · T= 0, (2.4)

where the superscript ext or int refers respectively to the suspending fluid or tothe capsule internal liquid. The jump of viscous traction across the membranes isq, the in-plane elastic tension tensor in the membrane is denoted T and ∇s is thesurface gradient operator. Equation (2.4) is the membrane equilibrium equation, whichexpresses the dynamic coupling between the solid membranes and the fluids.

As this fluid–structure interaction problem is now classical, we will only presentthe main hypotheses used here and refer the reader to the comprehensive review of

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Minuet motion of two capsules 892 A19-5

Barthès-Biesel (2016). The fluid velocity at any point x is written as a boundaryintegral on the surfaces S1 and S2 of the two capsules (Lac et al. 2007)

v(x)= v∞(x)−1

8πµ

∫S1∪S2

J(x, y) · q(y) dS(y), (2.5)

where J is the free space Green’s function given by

J(x, y)=I

‖x− y‖+(x− y)⊗ (x− y)‖x− y‖3

, (2.6)

where I is the identity tensor. The pressure p− p0 at a point x is given by

p(x)− p0 =−1

4π

∫S1∪S2

q(y) · (x− y)‖x− y‖3

dS(y), (2.7)

where p0 denotes the far field pressure.The capsule membrane is assumed to be an infinitely thin sheet of a three-

dimensional isotropic volume incompressible material that satisfies a neo-Hookean(NH) constitutive law. Bending resistance is neglected. The membrane constitutivelaw relates the principal elastic tensions (forces per unit arclength measured in themembrane plane) T1 and T2 to the two principal extension ratios λ1 and λ2

T1 =Gs

λ1λ2

[λ2

1 −1

(λ1λ2)2

], (2.8)

with a similar expression for T2, where the subscripts 1 and 2 are permuted. Forsimplicity, we have denoted these principal directions 1 and 2, but they should notbe confused with the Cartesian directions in space. The surface shear elastic modulusGs and area dilation modulus Ks are related by Ks = 3Gs (Barthès-Biesel 2016). Thecorresponding total deformation energy of the membrane of one capsule is given by

W =Gs

2

∫S

(λ2

1 + λ23 − 3+

1λ2

1λ22

)dS, (2.9)

where S stands for either S1 or S2.The main parameters are the capillary number

Ca=µγ aGs

, (2.10)

which measures the relative stiffness of the capsule, and the initial position of G2 attime t= 0, when the flow is suddenly started

X(0)=X(0)= {X(0)

1 , X(0)2 , X(0)

3 }. (2.11)

We shall discuss the typical case where X(0)1 and X(0)

3 are negative while X(0)2 is positive:

the flow of G2 occurs from left to right in the trajectory figures. The case X(0)3 > 0

provides the same results as the typical case since it corresponds to the symmetricconfiguration with respect to the shear plane. Positive values of X(0)

1 and negativeones for X(0)

2 correspond to the mirror image of the typical case, with respect to thex2x3-plane. When the crossing process is completed, the final steady position of G2 is{X( f )

1 , X( f )2 , X( f )

3 }. We define the final trajectory shifts of the capsule

δ2 = |X( f )2 | − |X

(0)2 |, δ3 = |X

( f )3 | − |X

(0)3 |, (2.12a,b)

which are computed for |X( f )1 | = 10a.

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892 A19-6 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

2.2. Numerical methodThe fluid–structure interaction problem is solved by means of the numerical schemethat couples a boundary integral method (BI) to solve the fluid flow and the finiteelement method (FE) to solve the membrane mechanics (Walter et al. 2010; Hu et al.2012). This method is well adapted to Stokes flows and has the advantage of requiringthe discretization of the capsule surfaces S1 and S2 only. It automatically accounts foran unbounded fluid domain. The model inputs are the capillary number Ca and theinitial position X(0) of capsule C2. Following Lac et al. (2007), we use the fact thatthe two capsules are identical to centre the flow field on the midpoint O of G1G2, sothat (2.5) can be solved on only one capsule and becomes

v(x)= v∞(x)−1

8πµ

∫S2

[J(x, y)− J(x,−y)] · q(y) dS(y). (2.13)

The capsule surface S2 is discretized using P2 triangle elements, in which 6 nodesare allocated at the vertices and the middle of each side. The mesh is generatedfrom the initial spherical shape by projecting a regular icosahedron to the sphere andthen subdividing each element subsequently to the desired precision. A mesh of 1280elements and 2562 nodes has been used in all the simulations, corresponding to acharacteristic mesh size 1hc =O(0.1a). Such a spatial discretization has been shownto lead to a relative error of order 10−3 on the Taylor deformation of a single capsulein shear flow (Walter et al. 2010). Furthermore, it can withstand some membranecompression without creating any numerical instability (Hu et al. 2012). The explicittime iteration is stable only if the time step is such that γ 1t<O(Ca1hc/a) (Walteret al. 2010). In the case of a NH membrane and Ca = 0.3, γ 1t = 5 × 10−4 allowsus to compute the capsule trajectory over long times without stability issues. We stopthe computation when the distance G1G2 is larger than 10a.

At time t = 0, C2 (positioned at {X(0)1 /2, X(0)

2 /2, X(0)3 /2}) and C1 (positioned at

{−X(0)1 /2,−X(0)

2 /2,−X(0)3 /2}) are subjected to the sudden start of the flow. The model

follows the motion of the capsule membrane over time. At any time, the model canthus output the position of the surface nodes, from which it is possible to infer, foreach capsule, the deformation and elastic tensions in the membrane as well as theposition and velocity of the centroid. The knowledge of this velocity allows us totranscript the results in the reference frame linked to C1.

The accuracy of the numerical model is checked by comparing capsules trajectorieswith those obtained by Lac et al. (2007) and Lac & Barthès-Biesel (2008), and alsoby assessing the influence of the time step and of the spatial discretization of thecapsule membranes (see the Appendix): the conclusion is that the centroid trajectoriesare accurate to within 0.05a.

3. Types of capsule interaction: simple crossing (leapfrog) or minuet motionA major finding of our work is that, depending on the initial position of C2, there

are two types of motion:

(i) Simple crossing (also denoted leapfrog): C2 catches up with C1, interacts andgoes away. The two capsules are not necessarily in the same shear plane.

(ii) Minuet motion: C2 catches up with C1, interacts, overpasses C1, reverses itsmotion and repeats the process one, two or three times before getting away.

The two motions are illustrated and analysed in the following.

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Minuet motion of two capsules 892 A19-7

20

-212

108

64

20

x3/a

x1/a

x 2/a

X 2/a

-2-4

-6-8

-10-12

20-2-4

Separationdirection t1

12840-4-8-12

2.0(b)(a)

1.5

1.0

0.5

0

-0.5

-1.0

X1/a

G1X(0)

X2(f)

X2(m)

X1(t1) = 0

x1/a x1/a

x 2/a x 2/a

210

-1-2-2-1 0 1 2 210-1-2

10-1

1(d)(c)

0

-1

x1/a

x 3/a

10-1

1(e)

0

-1

x3/a

x 2/a

10-1

1(f)

0

-1 SingleC1(t1)C2(t1)

x3/a

FIGURE 2. Leapfrog motion of two capsules with their centres in the same shear plane(X(0)

1 /a=−10, X(0)2 = 0, X(0)

3 = 0, Ca= 0.3). (a) Three-dimensional view of the trajectoryof C2. (b) G2 trajectory in the shear plane. (c) Three-dimensional view of the deformedcapsules at t1 when X1(t1) = 0. (d–f ) C1 intersections with the three coordinates planesat t1.

3.1. Single interaction: leapfrog motionAs a reference, we consider the situation where the two capsules are in the shear planeon the same streamline (X(0)

1 = 10a, X(0)2 = 0, X(0)

3 = 0, Ca= 0.3). Similar computationshave been made by Lac et al. (2007) for pre-inflated capsules, but only the value ofthe final trajectory shift is reported. Under Stokes flow conditions, the capsules mustremain in this plane. The trajectory of C2 is shown in figure 2(a,b) (movie 1 availableat https://doi.org/10.1017/jfm.2020.181).

When the flow is started, the two capsules are far enough from each other that theybehave as if they were almost alone in the fluid. Within γ t∼ 6 they reach a roughlyellipsoidal shape around which the membrane rotates, as shown in figure 2(d) (seethe review by Barthès-Biesel (2016)). The deformation and rotation of C1 lead to astresslet that creates a small velocity field and a small depression. This perturbationdisplaces C2 along the velocity gradient in the direction that moves it towards C1, asshown in figure 2(b). Note that this phenomenon is the opposite of the one observedby Doddi & Bagchi (2008) when inertia is taken into account. As a consequence,V2(t) > 0 for t > 0 and C2 is convected towards C1, as shown in figure 2(a,b). Theonly way for C2 to pass C1 is to ‘jump’ over it in the x1x2-plane (thus the term‘leapfrog motion’). The fact that the motion is constrained to the shear plane leadsto a high velocity difference ‖V − v∞‖/γ a in the x1- and x2-directions (figure 3).The evolution of the pressure at the midpoint between G1 and G2 (2.7) can alsoexplain the interaction phenomenon: indeed, as |X1(t)/a| decreases, the pressure in

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892 A19-8 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

129630-3-6-9-12

1.5

1.2

0.9

0.6

0.3

0

-0.3

-0.6

-0.9

X1/a129630-3-6-9-12

X1/a

V 1/©�

a

V 3/©�

aV 2

/©�a

0.40.30.20.1

0-0.1-0.2

0.10

0.05

0

-0.05

-0.10

LeapfrogMinuet√∞

FIGURE 3. Evolution of C2 velocity during a leapfrog (X(0)1 /a=−10, X(0)

2 = X(0)3 = 0) or

minuet (X(0)1 /a= X(0)

3 /a=−2.6, X(0)2 = 0) for Ca= 0.3.

12840X1/a

-4-8-12 12840X1/a

-4-8-12

1.2(a) (b)0.9

0.6

0.3

0

-0.3

-0.6

-0.9

-1.2

(p −

p0)

/(G

s/a)

(W −

W∞

)/G

sa2

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

LeapfrogMinuet

Initial state

G1X(0)

FIGURE 4. Pressure and membrane energy evolution during the leapfrog and minuetmotions for Ca= 0.3. (a) Pressure p at the mid-point between G1 and G2; (b) membraneelastic energy W. Leapfrog: X(0)

1 /a = −10, X(0)2 = 0, X(0)

3 = 0. Minuet: X(0)1 /a = −2.6,

X(0)2 = 0, X(0)

3 /a=−2.6.

the lubrication film between the two capsules increases (figure 4a) and pushes C2

in the x2-direction, along which the maximum displacement of G2 is X(m)2 . As C2

overtakes C1, the widening of the lubrication film leads to a depression (figure 4a),which decreases X2, until a final steady value X( f )

2 is reached when the two capsulesare far apart (|X1| > 6a, in this case). The final trajectory shift δ2 = |X

( f )2 − X(0)

2 | is0.9a, and is equal to the one found by Lac et al. under the same flow situation fora 5 % pre-inflation. The finite value of δ2 indicates that the two capsule interactionleads to self-diffusion effects in a dilute suspension of capsules. The deformedprofiles of C1 (equivalently of C2) in the x1x2-, x1x3- and x2x3-planes are shown infigure 2(d–f ) at time γ t1 = 9.7, when the two capsules cross, which we define byX1(t1) = 0. Figure 2(d, f ) shows that there is indeed a thin lubrication film between

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020.

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Minuet motion of two capsules 892 A19-9

1210

86

42

0-2

-4-6

-8-10

-12

x1/a

x1/a

20-2

-4 x3/a

x3/a

20

-2

x 2/a x 2

/a

20-2-4-202

20

-2

t2

t2(a) (b)

x1/a x3/a

x 2/a

20-2-4-202

20

-2

t1

t1

Separationdirection

12840-4-8-12X1/a

X 2/a

X 3/a

0.60.40.2

0-0.2-0.4-0.6-0.8

10

-1-2-3-4

G1

G1

t2

t1

t1

t2

X(0)

X(0)

X2(tr1) = 0

X2(tr1) = 0X2

(f)

X2(m1)

X2(m2)

10-1x1/a

x 2/a

1(c)

0

-1

10-1x1/a

x 3/a

1(d)

0

-1

10-1x3/a

x 2/a

1(e)

0

-1 SingleC1(t1)C1(t2)

FIGURE 5. Minuet of two capsules (X(0)1 /a = −2.6, X(0)

2 = 0, X(0)3 /a = −2.6, Ca = 0.3).

(a) Three-dimensional view of the trajectory of C2. (b) Projections of G2 trajectories inthe x1x2- and x1x3-planes. (c–e) C1 profile intersections with the three coordinates planesat times t1 and t2 when X1(t1)= X1(t2)= 0.

the two capsules (which corroborates the pressure build-up) and that the two capsulesare highly deformed. The global deformation can be assessed through the membraneelastic energy W given by (2.9). The value of W(X1) − W∞, where W∞ is thedeformation energy of a single capsule, allows us to estimate the intensity of themechanical interaction between the two capsules. During the close interaction, thetwo capsules undergo large transient deformation, leading to a peak in W − W∞(figure 4b). The separation process leads to some deformation oscillations, until afinal steady state is reached, which is identical to the single capsule one, elasticenergy wise (figure 4b).

3.2. Minuet motion

We now consider the case X(0)1 /a = −2.6, X(0)

2 = 0, X(0)3 /a = −2.6, Ca = 0.3, where

capsule C2 is located off the shear plane and can thus move freely in space: this leadsto trajectories that are completely different from the ones described above. As in theprevious case, the rotation of C1 displaces G2 along the velocity gradient, so that C2 isconvected towards C1. The three-dimensional trajectory of C2 is shown in figure 5(a),where the two capsules are shown in their initial positions (movie 2). In order toget a clearer grasp of the process, we analyse the trajectories of the projections of

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892 A19-10 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

G2 in the x1x2- and x1x3-planes in figure 5(b). As |X1(t)/a| decreases, the pressureincreases slightly (figure 4a): this leads to a slight increase in both |X2(t)| and |X3(t)|(figure 5b), which allows enough space for capsule C2 to pass C1 by moving aroundthe x2-axis (insets in figure 5a). Correspondingly, the maximum displacement |X(m1)

2 | issmaller than in the leapfrog situation. Similarly, the velocity difference ‖V− v∞‖/γ aremains small while occurring in all three directions, as shown in figure 3. The energyvariation W − W∞ is also very small (figure 4b), which indicates that there is littlemechanical interaction between the capsules. This point is further corroborated by thequasi-superposition of the deformed profiles of C1 at time t1 when X1(t1) = 0, andwhen it is alone in the flow (figure 5c–e).

As the capsules separate, the small depression (figure 4a) leads to a negativedisplacement along the x2-axis, which takes G2 into the reverse flow region andentices C2 to move back towards C1. The pressure is still negative when the reversaltakes place, so that |X3(t)| decreases. As a consequence, there is not enough space forC2 to move around C1 in a x1x3-plane and a sideways leapfrog motion takes place,which is qualitatively similar to the one that occurs in the shear plane, as describedin the previous section: there is a significant pressure build-up in the lubrication film,followed by a depression as the capsules part (figure 4a). The pressure variationleads to an increase of |X2(t)| up to a value |X(m2)

2 |, which is large enough to allowthe further decrease to the final displacement |X( f 2)

2 |, without crossing into a reverseflow region. At time t2 when X1(t2) = 0, the two capsules are closer than at timet1 and thus undergo a transient deformation, as appears in figure 5(a) (inset) and infigure 5(d,e). Correspondingly, the mechanical energy W −W∞ undergoes a transientvariation, which, however, is much smaller than the one that occurs when the capsulescross in the shear plane (figure 4b). We deduce that minuet motion is less energyconsuming than leapfrog motion.

If we now start with the same initial conditions (X(0)1 /a=−2.6, X(0)

2 = 0, X(0)3 /a=

−2.6) but reduce the capsule deformability by setting Ca= 0.2, two reversals occur,as shown in figure 6 (movies 3 and 4). The first reversal is essentially the sameas for Ca = 0.3. However, when the capsules cross again at time t2, there is stillenough space to allow crossing around the x3-axis. Furthermore, the displacementalong the x2-axis is still small (X2(t2)/a = −0.39), compared to the one observedfor Ca = 0.3 (X2(t2)/a = −0.51), so that the pressure induced trajectory shift forcesG2 to cross again into the reverse flow region: the capsule is thus convected againtowards C1. During the third crossing, since |X3(t3)|/a < 2, the capsule has to do asideways leapfrog motion that leads to a displacement X2(t2)/a = 0.92, that is largeenough to accommodate the trajectory shift without changing the flow direction of thecapsule: the latter finally goes back in the direction where it came from. Note that, attime t3, the two capsules undergo a significant transient deformation, due to a stronginteraction. This phenomenon is unexpected in view of the fairly large initial distancebetween the capsules.

4. What factors determine the motion type?The type of motion (leapfrog or minuet) is determined by the evolution of X2(t)

after crossing, since it is a change of sign of X2(t) that causes reversal of motion. Atcrossing, the film pressure displaces G2 along the velocity gradient, to a maximumvalue |X(m)

2 |. After crossing, the separation process and the subsequent depression leadto a displacement ∆2 of G2 back to the x1-axis. For leapfrog motion, the trajectoryshift is easily identified as ∆2=|X

(m)2 −X( f )

2 | (see figure 2b). When reversal occurs, the

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Minuet motion of two capsules 892 A19-11

0-2-4-6

-4-2

02

46

8

210

-1-2

x 2/a

x1/ax3/a

Separationdirection

Shear plane

t3t1

t2

(a)

12840-4-8-12X1/a

1.21.00.80.60.40.2

0-0.2-0.4-0.6

X 2/a

X 3/a

G1

G1

t1

t1

t2

t3

t3

t2

X(0)

X(0)

tr1

tr1

tr2

tr2

X2(f)

X2(m3)

X2(m1)

X2(m2)

1(b)

(c)

0-1-2-3-4

x1 /a x3/a

2

(d)

10

-1-2-2

-10

12

-4-3-2-1012

x 2/a

x1 /a x3/a

2

(e)

10

-1-2-2

-10

12

-4-3-2-1012

x 2/a

x1 /a x3/a

2

(f)

10

-1-2-2

-10

12

-4-3-2-1012

x 2/a

t1 t2 t3

FIGURE 6. Minuet with multiple reversals for X(0)1 /a = −2.6, X(0)

2 = 0, X(0)3 /a = −2.6,

Ca = 0.2. The instants of close interaction when X1 = 0 are denoted t1, t2 and t3 inchronological order. (a) Three-dimensional view of the trajectory of C2. (b,c) Projectionsof G2 trajectories in the x1x2- and x1x3-planes. (d) Three-dimensional profiles at closeinteraction.

film pressure decrease also leads to a trajectory shift ∆2, which is difficult to evaluatein a simple fashion. However, since ∆2 is a consequence of the separation process, wecan surmise that it follows an evolution similar to the one found in leapfrog motion inthe shear plane, i.e. that it decreases when the capsule separation |X(0)

2 | and/or |X(0)3 |

increase and when the capsule deformability Ca increases (Lac et al. 2007; Lac &Barthès-Biesel 2008; Pranay et al. 2010; Omori et al. 2013; Gires et al. 2014). Weconclude that, whenever |X(m)

2 | is less than ∆2, reverse motion is to be expected. Themain parameters that determine the values of |X(m)

2 | and of ∆2, are the initial capsuleseparation X(0) and the capillary number Ca. We now study their influence separately.

4.1. Effect of the initial capsule separation X(0)

The effect of the initial offset |X(0)3 | from the shear plane is shown for Ca = 0.3,

X(0)1 /a = −3.0 and X(0)

2 /a = 0 in figure 7. When |X(0)3 | is small (e.g. |X(0)

3 /a| 6 2),the situation is close to the one when the two capsules are in the same shear plane.Correspondingly, they undergo a sideways leapfrog motion with a displacement

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892 A19-12 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

76543210-1-2-3-4-5-6-7X1/a

X 3/a

X 2/a

2(b)

(a)

1

0

-1

-2

-3

-4

-5

|X3(0)| increases

X3(0)/a = 0

X3(0)/a = -1.0

X3(0)/a = -2.0

X3(0)/a = -2.6

X3(0)/a = -3.0

X3(0)/a = -4.0

2

1

0

-1

-2

FIGURE 7. Effect of the initial position X(0)3 on the capsule trajectory for Ca = 0.3,

X(0)1 /a=−3.0 and X(0)

2 /a= 0. Projections of G2 trajectories in the x1x2-plane (a) and inthe x1x3-plane (b). When |X(0)

3 |/a 6 2, capsule C2 has not enough room to move aroundC1: it overpasses it with a sideways leapfrog motion.

|X(m)2 /a| > 0.4, which is large enough to allow direct crossing (figure 7a). For

a larger offset |X(0)3 /a| > 2.6, C2 has room to move around C1 in an x1x3-plane:

then the displacement |X(m)2 /a| ∼ 0.2 is small (figure 7a), and thus reversal motion

occurs during separation. As |X(0)3 | increases, the influence of C1 decreases and

the two capsules have almost no relative velocity. The motion shown in figure 7 for|X(0)

3 /a| = 4 is near the limit of what can be reasonably computed: indeed, the capsulereaches X1 = 0 at time γ t1 = 50 at the first crossing, and at time γ t2 = 266 at thesecond crossing. We conclude that the minuet motion is slow.

We now turn to the effect of the initial distance |X(0)1 | on the trajectory of C2,

as shown in figure 8 for Ca = 0.3, X(0)2 /a = 0 and X(0)

3 /a = −2.6. Note that, since|X(0)

3 /a|> 2, C2 has room to move around C1 in an x1x3-plane. However, the far fieldperturbation created by C1 is a stresslet, which varies as the square of the inversedistance G1G2. It is this perturbation that displaces G2 along the x2-axis and gives C2

the small relative approach velocity, which leads to crossing. This perturbation velocityvaries as [X(0)

1 /a]−2

when |X(0)1 |/a � 1 (|X(0)

1 |/a > 4 in this particular case). Eventhough it is small, its prolonged effect over a long time leads to a significant |X(m)

2 |

displacement and thus to a sideways leapfrog motion. Note that the initial positionX(0)

2 /a= 0 is unstable when the capsules interact. However, when the initial distance|X(0)

1 | becomes of order a (e.g. |X(0)1 |/a = 3), the displacement |X(m)

2 | has no time tobuild up and remains small: then reversal occurs.

Similar situations of capsule interaction have been considered by Lac & Barthès-Biesel (2008), who did not report any motion reversal, even when they studied three-

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Minuet motion of two capsules 892 A19-13

76543210-1-2-3-4-5-6-7X1/a

X 3/a

X 2/a

(b)

(a)

2

1

0

-1

-2

-3

-4|X1

(0)| increases

X1(0)/a = -2.6

X1(0)/a = -3.0

X1(0)/a = -4.0

X1(0)/a = -6.0

2

1

0

-1

-2 Close trajectories

FIGURE 8. Effect of the initial position X(0)1 on the capsule trajectory for Ca = 0.3,

X(0)2 /a=0 and X(0)

3 /a=−2.6. Projections of G2 trajectories in the x1x2-plane (a) and in thex1x3-plane (b). When |X(0)

1 |/a> 4, capsule C2 has moved sufficiently across the streamlinesthat it can overpass C1 with a sideways leapfrog motion.

dimensional motions with non-zero values of X(0)3 . This is due to the fact that they

started with X(0)2 /a > 0.5 and X(0)

1 /a > 10, values which were too large for minuet tooccur.

4.2. Effect of capsule deformabilityUnder given flow conditions, the capsule deformability is accounted for by thecapillary number Ca and increases with it. Some global results are presented in theform of phase diagrams. For X(0)

2 = 0 and |X(0)1 | = |X

(0)3 |, the effect of varying the

initial capsule separation and deformability is shown in figure 9(a). Minuet thusoccurs approximately for 2< |X(0)

3 |/a< 4. For |X(0)3 |/a> 4∼ 5, the capsule separation

is so large that the relative velocity is very small and the capsule doublet configurationremains essentially stationary. The effect of Ca is complex: for a typical separation|X(0)

3 |/a= 2.6 and up to Ca6 0.7, a minuet takes place with one reversal for moderatevalues of deformability (0.2 < Ca < 0.7) or two (or more) reversals when Ca 6 0.2.This transition from one to two reversals when Ca is reduced has been illustratedin § 3.2. However, for large capsule deformability (Ca > 0.7), no motion reversaloccurs: the capsules are so deformed and tilted towards the flow direction, that, whenthey overpass, their trajectory perturbation is small. The effect of X(0)

2 is shown infigure 9(b) for the typical case |X(0)

1 |/a= |X(0)3 |/a= 2.6. Minuet occurs only for small

values of |X(0)2 |/a and moderate Ca, i.e. for small relative velocities between two

capsules with moderate deformability. This limited range of minuet motion explainswhy it had not been detected before.

Note that the inherent deformability of a capsule, even when Ca is very small,makes it different from a rigid sphere. Consequently, it is impossible to find permanent

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892 A19-14 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

0.80.70.60.5Ca Ca

0.40.30.20.1

3.4(a) (b)

3.2

3.0

2.8

2.6

2.4

2.2

Minuet

Minuet

Leapfrog Leapfrog

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.20

0.15

0.10

0.05

0

|X3(0

) |/a

|X2(0

) |/a

FIGURE 9. Phase diagrams for motion type: (a) motion type as a function of Ca andX(0)

3 for X(0)1 = X(0)

3 and X(0)2 = 0. The points represent the positions of G2 at rest: × one

oscillation, + two oscillations,f three oscillations. (b) Motion type as a function of Caand X(0)

2 for |X(0)1 |/a= |X

(0)3 |/a= 2.6.

capsule doublets like those predicted by Batchelor & Green (1972b) for two spheresfreely suspended in simple shear flow: indeed, such doublets can occur only forperfect spheres, as pointed out by the authors.

4.3. Region of minuet motion

The domain around C1, where long term doublets are formed, is illustrated in figure 10for Ca = 0.3. The sphere of radius 2.2a centred on G1 is an exclusion region, thatleaves a space 0.2a between the two capsules. When X(0)

1 = X(0)2 = 0, the capsule

doublet separated by X(0)3 , remains stationary. Whenever the centre of capsule C2 is

located in the yellow area, the two capsules will cross once (leapfrog motion) andseparate. In the green area, the two capsules will remain close and oscillate a fewtimes before eventually separating. For |X(0)

3 |/a & 4, the interaction becomes weak,so that the doublet configuration remains essentially stationary. The minuet domainshown in figure 10 is in fact three-dimensional and extends in the x2-direction over asmall distance of order 0.06a for Ca= 0.3 (see figure 9b). When Ca is decreased, theboundary between leapfrog and minuet motions does not change appreciatively, butthe thickness of the three-dimensional minuet domain increases. Conversely, as shownin figure 10, when Ca increases, the boundary is tilted towards the x3-axis and theminuet domain thickness decreases. Note that for X(0)

2 > 0.5a, the whole region wouldbe yellow (apart from the exclusion area).

An important consequence of the minuet motion is linked to the fact that it tendsto push together two capsules that were initially distant and not expected to interactmuch. This point is illustrated in figure 6 where the initial separation G1G2−2a=1.7ais decreased during crossing to a film with thickness approximately 0.2a, which is thinenough to lead to potential physico-chemical interactions or damage.

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Minuet motion of two capsules 892 A19-15

43210-1-2-3-4

5

4

3

2

1

0

-1

-2

-3

-4

-5↓ Weak interaction ↓

↑ Weak interaction ↑

X1(0)/a

X 3(0) /a

FIGURE 10. Domain of doublet formation around capsule C1 for X(0)2 = 0, Ca= 0.3 and

a NH membrane. The sphere of radius 2a is a steric exclusion area, while the grey zonebetween the spheres of radii 2.2a and 2a is an exclusion region, that leaves a space 0.2abetween the two capsules. The points represent the positions of G2 at rest, and the symbolsindicate the motion type: × minuet, @ leapfrog, 6 steady doublet. The position of theboundary between the two domains for Ca= 0.5 is indicated by a dash line.

4.4. Effect of the membrane constitutive equationIt is of interest to assess the influence of the wall constitutive law, as all the aboveresults have been obtained for a neo-Hookean membrane. In particular, it is possibleto assume that the principal tensions and elongations are related by the Skalak law(SK) (Skalak et al. 1973), which reads

T1 =Gs

λ1λ2[λ2

1(λ21 − 1)+Cλ2

1λ22(λ

21λ

22 − 1)]. (4.1)

The surface shear elastic modulus is Gs and the area dilation modulus is givenby Ks = (1 + 2C)Gs. The two laws (2.8) and (4.1) predict the same smalldeformation behaviour for C = 1, but for large deformations, the NH law is strainsoftening, whereas SK law is strain hardening (Barthès-Biesel, Diaz & Dhenin 2002).Correspondingly, for the same value of Ca, the deformation of a single capsule insimple shear flow is larger for a NH membrane than for a SK one (Barthès-Biesel2016).

We can then expect that the transition between a leapfrog and a minuet motion willhappen for values of Ca that will be different for capsules with SK or NH membranes.In order to verify this prediction, we model the interaction of two capsules enclosedeither by a SK membrane (C = 1) or a NH membrane, in the case X(0)

1 /a = −3.0,X(0)

2 = 0, X(0)3 /a = −2.6 and Ca = 0.5. As shown in figure 11(a), the NH capsules

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892 A19-16 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

X1/a

X 2/a

210-1

2

1

0

-1

-2

NHSK

12840X1/a

X 2/a

X 2/a

-4-8-12

0.60.40.2

0-0.2-0.4-0.6-0.8

1(a)(b)0

-1-2-3-4

NHSK

G1

G1

t1

t1

X(0)

X(0)

FIGURE 11. Interaction process of two capsules with a NH membrane or a SK membranefor Ca= 0.5, X(0)

1 /a=−3, X(0)2 = 0, X(0)

3 /a=−2.6. (a) The NH capsule does a sidewaysleapfrog motion, while the SK capsule does a motion reversal. (b) Deformed profiles ofthe capsule C1 at time t1 defined by X1(t1)= 0.

do a sideways leapfrog motion whereas the SK capsules undergo one oscillation andreverse their direction of motion. The explanation for this difference of behaviour islinked to the fact that the NH capsule is more deformed than the SK one (figure 11b),and that its trajectory displacement ∆2 is thus small enough to prevent it from goinginto the reverse flow region.

5. Global effects: trajectory shift and doublet duration5.1. Trajectory shift

When two capsules with a NH membrane are in the same shear plane (X(0)3 /a=0), it is

a well-established fact that, after crossing, the two capsules are irreversibly displacedfrom their initial trajectory: for a given value of Ca, the trajectory shift δ2 = |X

( f )2 | −

|X(0)2 | is maximum for X(0)

2 /a= 0, decreases when |X(0)2 | increases and becomes almost

zero for |X(0)2 |/a > 2 (Lac & Barthès-Biesel 2008; Pranay et al. 2010; Omori et al.

2013; Gires et al. 2014). The effect of X(0)3 /a is mostly reported for the case X(0)

2 /a=0.5: then δ2 is maximum for X(0)

3 /a = 0, decreasing to almost zero for |X(0)3 /a > 2

(Lac & Barthès-Biesel 2008; Gires et al. 2014). The trajectory shift along the vorticitydirection δ3= |X

( f )3 | − |X

(0)3 | is small and less than 0.1a (see also figure 15). The effect

of Ca is small and does not change the findings. The influence of the membraneconstitutive law on the leapfrog motion in the shear plane was studied by Pranay et al.(2010), who compared the effect of a NH or SK law (C= 10) on the trajectory shiftδ2. They found that, for the same Ca, δ2 is larger for a SK law than for a NH one:this is due to the high apparent rigidity of the SK law, linked to a high value of thearea dilation modulus.

The new results for X(0)2 /a= 0 are illustrated for Ca= 0.3 in figure 12, where they

are compared with the results of Lac & Barthès-Biesel (2008), obtained for X(0)2 /a=

0.5. For X(0)3 /a6 2 when a leapfrog motion occurs, the shift δ2 decreases with X(0)

3 /a

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Minuet motion of two capsules 892 A19-17

0 1 2 3 4 5

1.0(a) (b)

0.8

0.6

0.4

0.2

∂ 2/a

∂ 3/a

0

-0.2

|X3(0)|/a

0 1 2 3 4 5

|X3(0)|/a

0.5

0

-0.5

-1.0

-1.5

-2.0

-2.5

FIGURE 12. Irreversible trajectory shifts δ2 and δ3, computed at |X( f )1 |/a = 10, as a

function of X(0)3 (X(0)

1 /a=−3.0, X(0)2 = 0, Ca= 0.3). Open symbols correspond to leapfrog

motion, filled ones to minuet motion. The dash-dot line corresponds to the results of Lac& Barthès-Biesel (2008), obtained for X(0)

1 /a=−10, X(0)2 /a= 0.5 and β = 1.05.

and is approximately 50 % larger for X(0)2 /a= 0 than for X(0)

2 /a= 0.5 (figure 12a): thisis in agreement with the previously reported evolution of δ2 with X(0)

2 /a and X(0)3 /a.

However, when the doublet motion evolves from leapfrog to minuet, a bifurcationtakes place for |X(0)

3 |/a = 2.1 ± 0.1: δ2 jumps to values that are approximately oneorder of magnitude larger than the ones that would be expected from the previousX(0)

2 /a= 0.5 results or from the prolongation of the leapfrog curve for δ2.Similarly, whereas the shift δ3/a remains small during the leapfrog motion, it

becomes large and negative for X(0)2 /a > 2 (figure 12b). This means that, at the end

of the interaction process, the capsule C2 is nearer the shear plane x1x2 than when theflow started. The minuet motion thus leads to a shift δ3 along the vorticity direction,which tends to reduce the initial distance between the two capsules, in oppositionto a diffusive effect. But since the capsule separation in the x3-direction is reduced,a sideways leapfrog motion takes place, with a resulting large value of δ2, and thusdiffusive effects along the shear gradient direction. Decreasing Ca does not changemuch the above results. However, when the capsule deformability increases, thebifurcation occurs for increasingly large values of |X(0)

3 |/a and is difficult to computeas the minuet becomes very slow (see § 5.2).

The influence of the membrane law is shown in figure 13: the SK capsule undergoesa leapfrog motion for |X(0)

3 |/a 6 2 and then makes a minuet for |X(0)3 |/a > 2.2. The

evolutions of δ2 are almost superimposed for a NH membrane (Ca = 0.3) and fora SK membrane (Ca = 0.5). When the SK capsule deformability increases to Ca =1.0, the transition between leapfrog and minuet motions occurs further away, around|X(0)

3 |/a ∼ 3.5. We can then conclude that, indeed, the effect of a strain hardeningmembrane would just be to shift the results to larger values of Ca, without changingthe essence of the interaction process.

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892 A19-18 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

∂ 2/a

|X3(0)|/a

543210

1.0

0.8

0.6

0.4

0.2

0

-0.2

FIGURE 13. Trajectory shift δ2 as a function of X(0)3 for different membranes constitutive

laws (X(0)1 /a = −3.0, X(0)

2 = 0). The solid line corresponds to a neo-Hookean membranewith Ca = 0.3 (figure 12). The dashed lines correspond to a Skalak membrane (C = 1);E Ca= 0.5,@ Ca= 1.0. Open symbols: leapfrog motion; filled symbols: minuet motion.

0100

101

102

103

1 2 3 4 5

|X3(0)|/a

©�t

©�t1©�t1©� t1

©� t2©� t2©� t2

SK, Ca = 1.0:SK, Ca = 0.5:NH, Ca = 0.3:

FIGURE 14. Effect of Ca, X(0)3 and constitutive law on the times γ t1 and γ t2 of first

and second crossings (X(0)1 /a=−3.0 and X(0)

2 /a= 0).

5.2. Doublet duration

The minuet motion, when it occurs, is very slow, as shown in figure 14 for X(0)1 /a=

−3.0 and X(0)2 /a= 0. The time γ t1, at which the first crossing occurs, increases with

the separation |X(0)3 |/a, but does not depend on Ca, the membrane law or the type

of motion (leapfrog or minuet). The fact that γ t1 increases with |X(0)3 |/a is due to

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Minuet motion of two capsules 892 A19-19

20151050x1/a

x 2/a

-5-10-15-20 20151050x1/a

-5-10-15-20

1.50(a)

x 3/a

(b)

1.25

1.00

0.75

0.50

0.25

0

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

FIGURE 15. Projection of G2 trajectories for β = 1.05, Ca = 0.45 and different initialpositions. Capsule C2 starts from X(0)

1 /a = −20, X(0)2 /a = 0.5 and different X(0)

3 . Thecomparison between the results of Lac & Barthès-Biesel (symbols) and the present results(lines) shows very good agreement.

the fact that the relative velocity of C2, which is initially zero, builds up from aflow perturbation (due to C1), the intensity of which decreases with the square of thedistance between the two capsules. The time γ t2 of the second crossing is quite largeand increases significantly with capsule deformability and distance.

This means that the two capsules remain close to each other for a long time, whiledrifting slowly. When |X(0)

3 |/a> 4, γ t2 becomes too large to be reliably computed (thevalues of γ t2 for Ca= 1.0 and a SK membrane are very high, and are included hereto illustrate the phenomenon).

6. Discussion and conclusionThe situation that we have studied pertains to a semi-dilute suspension of capsules

that is suddenly put in motion by a simple shear flow. The novel aspect of our workis that we consider a pair of nearby capsules with their centres in (or near) theplane normal to the velocity gradient (X(0)

2 ∼ 0). This capsule configuration had neverbeen studied before, because it entails long computations. Indeed, the only resultson capsule interaction off the shear plane had been obtained for capsules with asignificant relative velocity, which prevented minuet from occurring and which led toweak capsule interactions and small trajectory displacement.

When X(0)2 ∼ 0, the important results are:

(i) The capsule pair can remain stable for a long time, while dancing a minuet.(ii) When they separate, the capsules can reverse direction.

(iii) When one or more oscillations occur, the irreversible trajectory shift is large.(iv) This minuet dance progressively leads the capsules to closely interact and deform

significantly.(v) The less deformable the capsule, the more prone it is to do a minuet.

(vi) The zone where a minuet can occur has been identified.

During the close interaction, the film thickness between the two capsules is ofthe order of ∼ 0.1a − 0.2a. For small capsules, this may lead to physico-chemicalinteraction.

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892 A19-20 X.-Q. Hu, X.-C. Lei, A.-V. Salsac and D. Barthès-Biesel

In order to make the problem consistent, we considered as an initial condition asuspension of spherical capsules at rest, and suddenly started the flow. When thecapsules are pre-deformed to the profile that they would have if they were alone inthe flow, we have verified that they take the same motion (leapfrog or minuet) as ifthey were initially spherical: the only difference is that, for pre-deformed capsules,the first close interaction happens γ t6 5 earlier than for spherical capsules. It followsthat the minuet interaction is not restricted to the transient start of the flow of asuspension. For example, from figure 6, we note that, at the end of the interaction,when C2 is located at X1/a= 5.0, X2/a= 0.54, X3/a=−1.1, it has almost recoveredits equilibrium shape. Suppose that C2 then meets a new capsule C3, located inthe vicinity of X1/a = 7.6, X2/a = 0.54, X3/a = 1.5. Being in the same relativeconfiguration as C1 and C2 were at time t= 0, capsules C2 and C3 will do a minuet.We can thus expect minuet motions in semi-dilute suspensions.

In conclusion, we have shown a novel and unexpected effect in the pair interactionof two capsules, that depends on the relative position of the two particles. It wouldbe interesting to check experimentally the existence of long lasting capsule doubletsthat do a minuet motion.

AcknowledgementsThis work was supported by the European Research Council (ERC) Consolidator

grant (MultiphysMicroCaps, no. 772191), by the National Natural Science Foundationof China (no. 11402084) and by the China Scholarship Council (Visiting ScholarScholarship of X.-Q.H.).

Declaration of interestsThe authors report no conflict of interest.

Supplementary moviesSupplementary movies are available at https://doi.org/10.1017/jfm.2020.181.

AppendixThe results of the BI–FE model (1280 elements) are first compared to those

obtained by Lac et al. (2007) and Lac & Barthès-Biesel (2008) for two pre-inflatedcapsules with a NH membrane and radius βa, where β is the inflation ratio. Thepre-stress is created by means of an internal pressure p0, which leads to an isotropicelastic tension T0 = p0a/2, given by Laplace’s law. For a neo-Hookean membrane,T0 = 6Gs(β − 1) in the limit of small inflation. The trajectories of G2 obtained withthe two methods show very good agreement, as illustrated in figure 15 for Ca= 0.45.

As the minuet motion is a novel phenomenon, which occurs over fairly longtimes, it is of importance to verify that the trajectories do not suffer from erroraccumulation over time. The test case consists of two capsules enclosed by a SKmembrane (C = 1) with Ca = 0.5, X(0)

1 /a = −3.0, X(0)2 = 0 and X(0)

3 /a = −2.4. Asshown in figure 13, those capsules undergo a minuet with one reversal. The choiceof a SK law rather than a NH one enables us to use larger values of Ca, and thuslarger time steps for similar trajectories. As shown in figure 16(a) for 1280 elementson the capsule surfaces, decreasing the time step γ 1t from 2× 10−3 to 5× 10−4 hasno effect on the trajectory. Conversely, we keep the same time step γ 1t = 2× 10−3

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Minuet motion of two capsules 892 A19-21

10.07.55.02.50-2.5-5.0-7.5-10.0X1/a

10.07.55.02.50-2.5-5.0-7.5-10.0X1/a

X 2/a

X 3/a

1.0

0.5

0

-0.5

-1.0

-1.5|dX2|/a = 0.03

|dX3|/a = 0.05

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

X 2/a

X 3/a

1.0(a) (b)

(c) (d)

0.5

0

-0.5

-1.0

-1.5

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.01050-5-10-15-20 1050-5-10-15-20

©�Ît = 2 ÷ 10-3

©�Ît = 1 ÷ 10-3

©�Ît = 5 ÷ 10-4

12805120

FIGURE 16. Effect of numerical parameters on the G2 trajectory, projected on the x1x2-and x1x3-planes for SK membranes, for Ca= 0.5, X(0)

1 /a=−3.0, X(0)2 /a= 0, X(0)

3 /a=−2.4.(a,b) Effect of time step for 1280 elements. (c,d) Effect of mesh refinement for γ 1t =2× 10−3.

and compare the trajectories obtained with two spatial meshes with 1280 or 5120elements, corresponding to mesh sizes 1hc =O(0.1a) and O(0.05a), respectively. Asshown in figure 16(b), the trajectories differ by at most one (fine) mesh size at theinteraction point. We conclude that the trajectories that are presented in this studyusing 1280 elements and γ 1t = 5 × 10−4, are reliable and that the final trajectoryshifts δ2/a and δ3/a have an error of the order of 0.05a.

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